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final_solutions - University of South Carolina Department...

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University of South Carolina Department of Physics and Astronomy Phys 706, Thermodynamics and Statistical Physics. Final Exam April 27, 2011 The test is open book/notes. You can use any materials you like. If you need extra paper or you forgot a calculator – please ask. There are 3 problems, each worth 10 points. It is enough to solve one problem to get a full credit. Choose any one you like. If you solve more, you will get extra credit. If you have a question about the problem’s formulation – please do not hesitate to ask. I will either answer your question or will tell you that I cannot do it without revealing the solution. Your solutions should have a readable form showing the flow of argument. You have to be able to express yourself in writing. Writing skills are important for any scientist. They are not just making your solutions look nicer. If you cannot explain yourself clearly, you probably do not understand what you are doing. You can still try to guess the answer for a problem. If the solution is a number or an expression, I will accept your answer (guessing will not be accepted for problems where the answer is “yes” or “no”). However, if no explanations are given, only correct answers will be graded positively. There will be no partial credit in the absence of correct, readable explanations. GOOD LUCK! 1 Ideal gas with gravity field You have a cylindrical container of height h and bottom area A filled with classical, ideal, single-atomic gas. The total number of atoms is N , the mass of each atom is m , and the temperature is T . Since every molecule is attracted to the Earth with the force mg , there will be more molecules at the bottom of the cylinder than at the top. (a)[2 points] Calculate the dependence of the gas concentration n ( z ) as a function of height z . (The uneven n has to create pressures which balance the gravity force acting on each element of gas.) (b)[8 points] In addition to the kinetic energy the gas has some potential energy in the gravity field. Find the total heat capacity of the gas in the cylinder. Find the limits at T → ∞ and T 0. Solution: (a) If you consider a thin horizothal slice of gas of between the planes z and z + dz , the difference between the pressures at its bottom and top has to be equal to the weight of the gas. This gives mgn ( z ) dz = P ( z ) - P ( z + dz ) = - dP dz dz
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Using P = nk B T we transform this to a differential equation dn dz = - mg k B T n ( z ) The solution is n ( z ) = n (0) e - ( mg/k B T ) z = n (0) e - z/z 0 where n (0) is the concentration at the bottom of the cylinder and z 0 k B T/mg is a characteristic length of concentration decay.
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